While the notion of quantum chaos is tied to random-matrix spectral correlations, also eigenstate properties in chaotic systems are often assumed to be described by random matrix theory. Analytic insights into such eigenstate properties can be obtained by the recently introduced partial spectral form factor, which captures correlations between eigenstates. Here, we study the partial spectral form factor in chaotic dual-unitary quantum circuits. We compute the latter for a finite connected subsystem in a brickwork circuit in the thermodynamic limit, i.e., for an infinite complement. For initial times, shorter than the subsystem’s size, spatial locality and (dual) unitarity implies constant partial spectral form factor, clearly deviating from the linear ramp of the partial spectral form factor in random matrix theory. In contrast, for larger times we prove, that the partial spectral form factor follows the random matrix result up to exponentially suppressed corrections. We supplement those exact results by semi-analytic computations performed directly in the thermodynamic limit.